Method and apparatus for reordering complex SQL queries containing inner and outer join operations

ABSTRACT

A method and apparatus for reordering complex SQL queries containing joins, outer and full outer joins. The method and apparatus first translates the query into a hypergraph representation. Required sets, conflict sets and preserved sets are then generated for the query hypergraph. Using the required sets, a plurality of plans are enumerated, wherein the plans represent associative re-orderings of relations in the query. SQL operators are selectively assigned to each of the enumerated plans using the conflict sets and/or preserved sets, so that the results from the plans are identical to the original query. A novel Modified General Outer Join (MGOJ) operator may be assigned to the root of a sub-tree, wherein the MGOJ operator is a compensation operator. The operator assignment is performed recursively for the root of each sub-tree in the plan. One of the enumerated plans (generally the most optimal) is then selected for execution.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of and commonly-assigned patent application Ser. No. 08/326,461, entitled "METHOD AND APPARATUS FOR REORDERING COMPLEX SQL QUERIES CONTAINING INNER AND OUTER JOIN OPERATIONS," filed on Oct. 20, 1994, by Gautam Bhargava et al., attorney's docket number ST994020A, now U.S. Pat. No. 5,680,603, which application is incorporated by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates in general to database management systems performed by computers, and in particular to method and apparatus for reordering complex SQL queries in a relational database management system.

2. Description of Related Art

Relational DataBase Management System (RDBMS) products using a Structured Query Language (SQL) interface are well known in the art. The SQL interface has evolved into a standard language for RDBMS products and has been adopted as such by both the American Nationals Standard Organization (ANSI) and the International Standards Organization (ISO). In RDBMS products, all data is externally structured into tables. The SQL interface allows users to formulate relational operations on the tables either interactively, in batch files, or embedded in host languages such as C, COBOL, etc. Operators are provided in SQL that allow the user to manipulate the data, wherein each operator operates on either one or two tables and produces a new table as a result. The power of SQL lies on its ability to link information from multiple tables or views together to perform complex sets of procedures with a single statement.

The current state-of-the-art in SQL query optimization has few solutions for optimizing query expressions involving joins, outer joins, and full outer joins. Several researchers have performed pioneering work in this area, as reflected in the following publications:

Galindo-Legaria, C., and Rosenthal, A., "How to extend a conventional optimizer to handle one- and two-sided outer joins", Proceedings of Data Engineering, pp. 402-409, 1992, (hereinafter referred to as " GALI92a!");

Galindo-Legaria, C. A., "Algebraic optimization of outer join queries", Ph.D. dissertation, Dept. of Applied Science, Harvard University, Cambridge, 1992, (hereinafter referred to as " GALI92b!"); and

Rosenthal, A. and Galindo-Legaria, C., "Query graphs, implementing trees, and freely-reorderable outer joins", SIGMOD, pp. 291-299, 1990, (hereinafter referred to as " ROSE90!").

However, there are numerous problems in the above prior art techniques. More specifically, GALI92a!, GALI92b!, and ROSE90! make a number of assumptions that severely restrict the application of their research to real-world RDBMS products.

One assumption made in the prior art is that there are no duplicates in base relations (an unstated assumption present in GALI92a!, GALI92b!, ROSE90!). Since most commercial RDBMS products allow duplicates in relations, the following identity in GALI92a! and GALI92b! does not hold in the presence of duplicates: ##EQU1## for relations r₁, r₂, and r₃, wherein the Generalized Outer Join (GOJ) is a specialized operator used for optimization. In the absence of this identity, GALI92b! would not be able to consider a significant number of re-orderings for queries containing outer joins.

Another assumption made in the prior art is that projection removes all duplicates. Most commercial RDBMS products, including the DB2 family of products provided by International Business Machines Corporation, the assignee of the present invention, support two kinds of projections: one that removes duplicates (SELECT DISTINCT) and another that does not (vanilla SELECT). Duplicate elimination (for that instance, selection predicates) may be specified on an intermediate result of a query. The intermediate result may be a view or table expression that contains a projection which eliminates duplicates, e.g., where DISTINCT is specified for the attributes.

Still another assumption made in the prior art is that outer join predicates refer to exactly two relations. If an outer join predicate references more than one attribute of a view or table expression, it is possible that these attributes may come from different base relations referenced in the view or table expression.

Thus, there is a need in the art for techniques for optimizing joins, one sided outer joins, and full outer joins in SQL queries.

SUMMARY OF THE INVENTION

To overcome the limitations in the prior art described above, and to overcome other limitations that will become apparent upon reading and understanding the present specification, the present invention discloses a method for reordering complex SQL queries containing joins, outer and full outer joins. An object of the present invention is to optimize such SQL queries, and thus improve their execution time. A feature of the invention is the identification and provision of an enhanced set of re-orderings for optimizing the SQL queries. Another feature of the invention is the provision of a Modified Generalized Outer Join (MGOJ) operation for facilitating the associative reordering of the SQL queries. This enhanced set of reorderings and the MGOJ operator are not recognized by existing optimizers available in prior art relational database management systems.

The present invention first translates the query into a hypergraph representation. Required sets, conflict sets and preserved sets are then generated for the query hypergraph. Using the required sets, a plurality of plans are enumerated, wherein the plans represent associative re-orderings of relations in the query. SQL operators are selectively assigned to each of the enumerated plans using the conflict sets and/or preserved sets, so that the results from the plans are identical to the original query. The novel MGOJ operator may be assigned to the root of a sub-tree, wherein the MGOJ operator is a compensation operator that facilitates the associative reordering of the query. The operator assignment is performed recursively for the root of each sub-tree in the plan. One of the enumerated plans (generally the most optimal) is then selected for execution.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers represent corresponding parts throughout:

FIG. 1 illustrates the computer hardware environment of the present invention;

FIG. 2 is a flowchart illustrating the steps necessary for the interpretation and execution of SQL statements in an interactive environment according to the present invention;

FIG. 3 is a flowchart illustrating the steps necessary for the interpretation and execution of SQL statements embedded in source code according to the present invention;

FIG. 4 is a flowchart illustrating the method of optimizing SQL queries of the present invention, wherein an optimal plan or schedule is generated for a given SQL query that may contain complex predicates and may reference relations containing duplicates;

FIG. 5 is a flowchart illustrating the method of translating SQL queries into hypergraphs according to the present invention;

FIG. 6 is a flowchart illustrating the method of generating required sets for SQL queries according to the present invention;

FIG. 7 is a flowchart illustrating the method of generating conflict sets for SQL queries according to the present invention;

FIG. 8 is a flowchart illustrating the method of generating preserved sets for SQL queries according to the present invention;

FIG. 9 is a flowchart illustrating the method of generating association trees for SQL queries according to the present invention;

FIGS. 10A, 10B, and 10C together are a flowchart illustrating the method of performing conflict-free operator assignments for the association trees according to the present invention; and

FIGS. 11A, 11B, and 11C together are a flowchart illustrating the method of performing general operator assignments for the association trees according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In the following description of the preferred embodiment, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration a specific embodiment in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural and functional changes may be made without departing from the scope of the present invention.

ENVIRONMENT

FIG. 1 illustrates an exemplary computer hardware environment that could be used with the present invention. In the exemplary environment, a computer system 102 is comprised of one or more processors connected to one or more electronic storage devices 104 and 106, such as disk drives, that store one or more databases.

Operators of the computer system 102 use a standard terminal interface 108, such as IMS/DB/DC, CICS, TSO, or other interface, to perform various search and retrieval functions, termed queries, against the databases, typically using Relational DataBase Management System (RDBMS) software that incorporates the Structured Query Language (SQL) standard. In the preferred embodiment of the present invention, the RDBMS software comprises the DB2 product offered by IBM for the MVS or OS/2 operating systems. Those skilled in the art will recognize, however, that the present invention has application to any RDBMS that uses SQL.

The DB2 architecture includes three major components: (1) the IMS Resource Lock Manager (IRLM) 110, the Systems Services module 112, and the Database Services module 114. The IRLM 110 handles locking services, because DB2 treats data as a shared resource, thereby allowing any number of users to access the same data simultaneously, and thus concurrency control is required to isolate users and to maintain data integrity. The Systems Services module 112 controls the overall DB2 execution environment, including managing log data sets 106, gathering statistics, handling startup and shutdown, and providing management support.

At the center of the DB2 architecture is the Database Services module 114. The Database Services module 114 contains several submodules, including the Relational Database System (RDS) 116, the Data Manager 118, the Buffer Manager 120 and other components 122 such as an SQL compiler/interpreter. These submodules support the functions of the SQL language, i.e., definition, access control, retrieval, and update of user and system data.

FIG. 2 is a flowchart illustrating the steps necessary for the interpretation and execution of SQL statements in an interactive environment according to the present invention. Block 202 represents the input of SQL statements into the computer from the user. Block 204 represents the step of compiling or interpreting the SQL statements. An optimization function within block 204 may reorder the SQL query in a manner described in more detail later in this specification. Block 206 represents the step of generating a compiled set of runtime structures called an application plan from the compiled SQL statements. Generally, the SQL statements received as input from the user specify only the data that the user wants, but not how to get to it. This step considers both the available access paths (indexes, sequential reads, etc.) and system held statistics on the data to be accessed (the size of the table, the number of distinct values in a particular column, etc.), to choose what it considers to be the most efficient access path for the query. Block 208 represents the execution of the application plan, and block 210 represents the output of the results of the application plan to the user.

FIG. 3 is a flowchart illustrating the steps necessary for the interpretation and execution of SQL statements embedded in source code according to the present invention. Block 302 represents program source code containing a host language (such as COBOL or C) and embedded SQL statements. The program source code is then input to a pre-compile step 304. There are two outputs from the pre-compile step 304: a modified source module 306 and a Database Request Module (DBRM) 308. The modified source module 306 contains host language calls to DB2, which the pre-compile step 304 inserts in place of SQL statements. The DBRM 308 consists of the SQL statements from the program source code 302. A compile and link-edit step 310 uses the modified source module 306 to produce a load module 312, while an optimize and bind step 314 uses the DBRM 308 to produce a compiled set of runtime structures for the application plan 316. As indicated above in conjunction with FIG. 2, the SQL statements from the program source code 302 specify only the data that the user wants, but not how to get to it. The optimize and bind step 314 may reorder the SQL query in a manner described in more detail later in this specification. Thereafter, the optimize and bind step 314 considers both the available access paths (indexes, sequential reads, etc.) and system held statistics on the data to be accessed (the size of the table, the number of distinct values in a particular column, etc.), to choose what it considers to be the most efficient access path for the query. The load module 312 and application plan 316 are then executed together at step 318.

OPTIMIZATION

The method of optimizing SQL queries of the present invention eliminates the restrictive assumptions of the prior art, thus making it possible to perform comprehensive reorderings of complex queries containing joins and outer joins in SQL based commercial RDBMS products.

The prior art's first restrictive assumption of no duplicates in relations and second restrictive assumption that projection removes all duplicates are eliminated by the introduction of a novel binary operator, termed the Modified Generalized Outer Join (MGOJ). The MGOJ operator is sufficiently powerful to replace the Generalized Outer Join operator (GOJ) proposed in GALI92b!.

The prior art's third restrictive assumption that outer join predicates refer to exactly two relations is eliminated by the introduction of a new, simple, yet powerful concept of "precedence sets". This concept is easily integrated into query optimizers that use dynamic programming to enumerate the various reorderings for the query, like the commercial RDBMS products DM2 and DB2/CS V2 offered by IBM, the assignee of the present invention.

Finally, a note should be made regarding operator position. Since unary operators like selection and projection can appear anywhere in queries, binary join operators may not be adjacent to each other in a given user query. The results from GALI92b! and as well results from testing by the inventors of the present invention require that the binary operators should be adjacent to each other. Methods exist to remove unary operators as shown by GALI92b! and: Pirahesh, H., Hellerstein, J. M. and Hasan, W., "Extensible/rule based query rewrite optimization in Starburst", SIGMOD, pp. 39-48, San Diego, Calif., June 1992, (hereafter referred to as " PIRA92!"). Henceforth, the present invention assumes that binary operators are adjacent to each other.

DEFINITIONS

Following are definitions for a number of terms used in SQL queries. These definitions are required for an understanding of later portions of the present specification.

Tuple

A tuple t is a mapping from a finite set of attributes, R∪V, to a set of atomic (possibly null) values, where R is a non-empty set of real attributes and V is a non-empty set of virtual attributes, R∩V=φ, and the mapping t maps at least one virtual attribute vεV to a non-null value. For an attribute set X, t X! represents the values associated with attributes X under the mapping t, where X.OR right.R∪V and X≠φ.

The motivation behind the distinction between real and virtual attributes is to emphasize the difference between the real attributes available for manipulation (to the user of the RDBMS) and the virtual attributes used (by the RDBMS) for bookkeeping only. The set of real attributes of a tuple is the same as the schema of the tuple in the traditional relational algebra. These attributes are accessible to users and can be referenced externally, e.g., in user queries, etc. On the other hand, virtual attributes are (at times) used to provide unique conceptional tuple-ids to tuples, and are not accessible to users and cannot be referenced externally.

Relation

A relation r is a triple <R, V, E> where R is a non-empty set of real attributes, V is a non-empty set of virtual attributes, and E, the extension of relation r, is a set of tuples such that: ##EQU2## In this definition, R∪V is called the schema of relation r. Predicate

A predicate p over a set of real attributes sch(p) called the schema of p, is a total mapping from tuples to the Boolean values {TRUE, FALSE}, where sch(p) is the minimum set of attributes such that for all tuples t₁ and t₂ : ##EQU3## For a tuple t with real schema R.OR left.sch(p), p(t) is TRUE if and only if (∀Aεsch(p)) (i.e., substitution of t A! for A in p causes it to evaluate to TRUE).

Null-intolerant

A predicate p is null-intolerant if p evaluates to FALSE for tuples undefined on one or more attributes in sch(p). More formally, p is null-intolerant if: ##EQU4## Throughout the remainder of the present specification, it is assumed that all predicates are null-intolerant.

ALGEBRAIC OPERATORS

Following are definitions for algebraic operators used in SQL queries. These definitions are required for an understanding of later portions of the present specification.

Relations

Let r=<R,V,E>, r₁ =<R₁,V₁,E₁ > and r₂ =<R₂,V₂,E₂ > denote relations such that R₁ ∪R₂ =φ and V₁ ∪V₂ =φ.

Projection

The projection, π^(a) _(x) (r), of relation r onto attributes X is the relation <X,V,E'> where X.OR right.R and: ##EQU5##

The π^(a) operator is a projection operator that does not remove "duplicates" in the real attributes part of the source expression. The superscript a in π^(a) denotes the fact that all the virtual attributes of the source expression are included in the virtual schema of the result expression. For ease of reading, the superscript a is omitted from π^(a) whenever there is no ambiguity, so it can be written simply as π.

The projection, π^(c) _(X).sbsb.R_(X).sbsb.V (r), of relation r on attributes X_(R) X_(V) is the relation <X_(R),X_(V),E'>, where X_(R) .OR right.R, X_(V) .OR right.V and: ##EQU6##

In contrast to π, π^(c) allows a choice in the selection of the virtual attributes from the source expression. This operation is needed for defining the "Modified Generalized Outer Join" operator defined hereinafter.

Delta-Projection

The delta-projection, δ_(X).sbsb.R_(X).sbsb.V (r), of relation r on attributes X_(R) X_(V) is the relation <X_(R) X_(V),V_(new),E'>, where X_(R) .OR right.R, X_(V) .OR right.V, and: ##EQU7## which is a new, unique value. The δ operator models the "SELECT DISTINCT . . . " construct of SQL which allows elimination of "duplicates" from a relation. The δ operator is called the distinct projection operator and produces a result relation which has distinct values on the attributes X_(R) X_(V) and a new virtual attribute. Note that the X_(V) part of the tuple in the result expression is of theoretical importance mainly, and results in BHAR94! discuss the optimal "realization" of virtual ids.

Selection

The selection, δ_(p) (r), of relation r on predicate p is the relation <R,V,E'>, where sch(p).OR right.R, and:

    E'={t|(tεE)Λp(t)}

Cross Product and Difference

The cross product, r₁ ×r₂, and difference, r₁ -r₂, of relations r₁ and r₂ are the relations <R₁ R₂, V₁ V₂, E₁ ×E₂ > and <R, V, E₁ -E₂ >, respectively.

Union and Outer Union

The union, r₁ ∪r₂, of relations r₁ and r₂ is the relation <R, V, E₁ ∪E₂ >. The outer union, ##EQU8## is the relation <R₁ ∪R₂, V₁ ∪V₂, E'>, where: ##EQU9## Note that rows in ##EQU10## are padded with nulls for those attributes that are not present either in relation r₁ or in relation r₂.

Additional Algebraic Operators

In the following definitions, it is assumed that if predicate p is associated with join/outer/full outer join of relations r₁ and r₂ then p=p₁ Λp₂ Λ . . . Λp_(m), where p_(i) is a predicate such that sch(p_(i))∩R₁ ≠φ and sch(p_(i))∩R₂ ≠φ, 1≦i≦m.

Join

The join, ##EQU11## of relations r₁ and r₂ is the relation <R₁ R₂,V₁ V₂,E'>, where:

    E'={t|tε(E.sub.1 ×E.sub.2)Λp(t)}

Anti-Join

The anti-join, ##EQU12## of relations r₁ and r₂ is the relation <R₁, V₁, E'>, where: ##EQU13## Left and Right Outer Joins

The left outer join, ##EQU14## is the relation <R₁ R₂, V₁ V₂, E'>, where: ##EQU15## Relation r₁ in the above definition is called the preserved relation and relation r₂ is called the null supplying relation. The right outer join, ##EQU16## can similarly be defined in which r₁ is the null supplying relation and r₂ is the preserved relation.

Full Outer join

The full outer join, ##EQU17## of relations r₁ and r₂ is the relation <R₁ R₂, V₁ V₂, E'>, where: ##EQU18##

EXPRESSION AND EXPRESSION TREES

The following provides a recursive definition of expressions.

1. If r=<R, V, E> is a relation, then r is an expression. Henceforth, the shorthand notation X will be used to represent the expression X=<R_(x), V_(x), E_(x) >,

2. If X=<R_(x), V_(x), E_(x) > is an expression, then π^(a) _(x), (e) is an expression, where X'.OR right.R_(x).

3. If X=<R_(x), V_(x), E_(x) > is an expression, then δ_(X).sbsb.R_(X).sbsb.V (e) is an expression, where X_(R) .OR right.R_(x) and X_(v) .OR right.V_(x).

4. If X=<R_(x), V_(x), E_(x) > is an expression, then σ_(p) (X) is an expression, where sch(p).OR right.R_(x).

5. If X=<R_(x), V_(x), E_(x) > and Y=<R_(y), V_(y), E_(y) > are expressions, then ##EQU19## is an expression, where: ##EQU20## and p is a predicate such that sch(p)∩R_(x) ≠φ and sch(p)∩R_(y) ≠φ. Note that MGOJ p, X', Y'!, which is defined further hereinafter, does not appear in user expressions and is generated only by the optimizer in the preferred embodiment of the present invention.

6. If X=<R_(x), V_(x), E_(x) > is an expression, then so is (X), where (X)=<R_(x), V_(x), E_(x) >. This is termed parenthesization, and is required due to the fact that some of the binary operations defined above are not fully associative. Parenthesization determines the evaluation order so that expressions can be evaluated unambiguously. However, whenever there is no ambiguity, parentheses will be dropped freely.

An expression can also be represented by a corresponding expression tree in which the inner nodes are the operators occurring in the expression and the leaves are the base relations referenced in the expression. Let denote one of the binary operators defined in the previous section, then an expression tree T with left sub-tree T₁, right sub-tree T_(r) and root is denoted by: ##EQU21## Henceforth, the two equivalent representations are used interchangeably.

Those skilled in the art will recognize thae application of methods claimed in this application to a layer class of expressions supported by SQL.

HYPERGRAPHS AND ASSOCIATION TREES

A query is represented by hypergraph, defined as follows.

Hypergraph

A hypergraph G is defined to be the pair <V,E>, where V is a non-empty set of nodes and E is the set of hyperedges, such that E is a mapping on non-empty subsets of V (i.e., E:2^(v) →2^(v))

As a notational convenience, for hyperedge e=(V₁,V₂)εE, where V₁,V₂ ε2^(v). V₁ is referred to as sourceHypernode(e) and V₂ is referred to as destHypernode(e). Further, whenever there is no distinction required between V₁ and V₂, then they are simply called hypernodes. If |V₁ |=|V₂ |=1, hyperedge e is referred to as simply an edge and the hypernodes V₁ and V₂ as simply nodes.

A hypergraph is used to represent a query, where the set of nodes of the hypergraph correspond to relations referenced in the query, and edge represents a join (inner, one-sided outer, or full outer) operation involving a predicate between the 2 nodes of the edge, and a hyperedge represents an outer join (full or one-sided) between the sets of relations in its sourcehypernode and destHypernode. To clarify, edges correspond to predicates involving exactly two relations, whereas hyperedges correspond to outer join operations involving predicates that reference more than 2 relations.

Path

A path in a hypergraph G=<V,E> from node v₁ to node v_(n) is an alternating sequence of nodes and edges, v₁ e₁ . . . V_(i) e_(i) . . . e_(n-1) V_(n) such that:

1. V_(j) εV for 1≦j≦n and e_(k) εE for 1≦k≦(n-1).

2. v_(i) ≠v_(j) for 1≦i, j≦n and i≠j.

3. e_(j) ≠e_(k) for 1≦j, k≦(n-1) and j≠k.

4. Either v_(j) εsourceHypernode (e_(j)) and v_(j+1) εdestHypernode (e_(j)), or v_(j) εdestHypernode (e_(j)) and v_(j+1) εsourceHypernode (e_(j)), for 1≦j≦(n-1)

v₁ is called the starting node of the path and v_(n) the ending node of the path.

Directed and Bi-directed (Hyper)edges

A (hyper)edge is directed if it represents a one-sided outer join operation in the query. Further, a (hyper)edge is bi-directed if it represents a full outer join operation in the query.

Induced Subgraph

A hypergraph G'=<V',E'> is an induced subgraph of graph G=<V,E> if V'.OR right.V, E'.OR right.E and:

    E'={e|e=<V.sub.1,V.sub.2 >εE, V.sub.1 .OR right.V', V.sub.2 .OR right.V'}

An induced subgraph G'=<V',E'> of G is denoted by G|_(v').

Association Tree

For a query hypergraph G=<V,E>, an association tree T is defined for G as a binary tree such that:

1. leaves(T), and no relation appears in more than one leaf.

2. For any sub-tree T' of T, G|_(leaves)(T') is connected where G|_(leaves)(T') is the induced subgraph of G.

3. For any sub-tree T'=(T₁ '·T_(r) ') of T, let E_(T') denote the set of all edges in E that connect leaves of T₁ ' with leaves of T_(r) ', then either V₁ .OR right.leaves(T'₁) and V₂ .OR right.leaves(T'_(r)) or V₂ .OR right.leaves(T'₁) and V₁ .OR right.leaves (T'_(r)).

MODIFIED GENERALIZED OUTER JOIN

In this section, the novel Modified Generalized Outer Join (MGOJ) operator is presented, which is used to reorder complex queries. Next, the procedures ASSOCIATION TREE, CONFLICT-FREE OPERATOR ASSIGNMENT and GENERAL OPERATOR ASSIGNMENT are presented that generate the optimal plan for a given query.

Consider the query: ##EQU22##

If only joins, outer and full outer joins are employed, then this query can only be executed in the manner in which it is specified. To re-order this query and other similar queries, the Generalized Outer Join (GOJ) operator was introduced in the publication: Dayal, Umeshwar, "Of nests and trees: A unified approach to processing queries that contain nested subqueries, aggregates, and quantifiers", VLDB, pp. 197-208, 1987 (hereinafter referred to as " DAYA87!").

The GOJ of relation X₁ by relation X₂ preserving X₃, where X₃ .OR right.X₁, is equal to: ##EQU23##

The publication GALI92a! identified a small error in this operator and modified it to: ##EQU24##

Since both these operators remove duplicates in the anti-join part, they cannot be used in the presence of duplicates. The present invention provides a new operator, the Modified Generalized Outer Join (MGOJ), that can handle duplicates in the relations.

In the following definition, let r₁ =<R₁,V₁,E₁ > and r₂ =<R₂,V2,E₂ > be two relations such that R₁ ∩R₂ =φ and V₁ ∩V₂ =φ. Further, let X_(i) =<R_(x).sbsb.i,V_(x).sbsb.i,E_(x).sbsb.i >, 1≦i≦n, and Y_(j) =<R_(y).sbsb.j,V_(y).sbsb.j,E_(y).sbsb.j >, 1≦j≦n', be relations such that R_(x).sbsb.i ∩R_(x).sbsb.k =V_(x).sbsb.i ∩V_(x).sbsb.k =φ. R_(y).sbsb.j ∩R_(y).sbsb.s =V_(y).sbsb.j ∩V_(y).sbsb.s =φ, R_(x).sbsb.i .OR right.R₁ and R_(y).sbsb.j .OR right.R₂, where i≠k, j≠s, 1≦i, k≦n and 1≦j, s≦n'.

The MGOJ operator, which is expressed in the form:

    r.sub.1 MGOJ p,X.sub.1, . . . ,X.sub.n,Y.sub.1, . . . ,Y.sub.n' !r.sub.2

of relations r₁ and r₂ while preserving relations X_(i) and Y_(j) is the relation <R₁ R₂,V₁ V₂,E'>, where 1≦i≦n, 1≦j≦n' and E' is given by: ##EQU25##

In the preferred embodiment, the MGOJ operator is available only to the optimizer during re-ordering of binary operations and cannot appear in user queries. However, those skilled in the art will recognize that such an operator could, in fact, become part of the SQL interface available to users.

GENERATION OF THE OPTIMAL QUERY

FIG. 4 is a flowchart illustrating the method of optimizing SQL queries of the present invention, wherein an optimal plan or schedule is generated for a given SQL query that may contain complex predicates and may reference relations containing duplicates. Block 402 represents the acceptance of the SQL query (either interactively from the user or extracted from program source code). Block 404 represents the translation of the query into a hypergraph. Block 406 represents the generation of the required sets, conflict sets, and preserved sets for the hypergraph. Block 408 represents the generation of all possible association trees for the query hypergraph. Block 410 represents the assignment of operators to the association trees to generate expression trees. Block 412 represents the selection of an optimal association tree (plan) for execution. Block 414 represents the execution of the reordered query against the relational database and the output of the results of the query to the user.

In the above method, the required sets and the conflict sets are generated independently of each other. Moreover, the association trees or plans and the conflict sets are also generated and enumerated independently of each other. Typically, however, the required sets and conflict sets are generated before the association trees or plans are enumerated. Moreover, the required sets, plans, and conflict sets are generated, enumerated, and generated before the operators are assigned.

HYPERGRAPH TRANSLATION

FIG. 5 is a flowchart illustrating the method of translating SQL queries into hypergraphs according to the present invention. Block 502 represents the input of the SQL query and the parsing of the query into its component parts. Block 504 represents the generation of a set of nodes corresponding to relations referenced in the query. Block 506 represents the generation of edges for all joins in the query, wherein each edge is generated between two of the nodes corresponding to a join operation involving a predicate between the two nodes. Block 506 also identifies an edge as being directed when it represents a one-sided outer join operation in the query. In addition, block 506 identifies an edge as being bi-directed when it represents a full outer join operation in the query. Block 508 represents the generation of hyperedges for all outer joins in the query, wherein each hyperedge is generated between two of the nodes corresponding to a left outer join, a right outer join, or a full outer join operation involving a predicate that references a plurality of relations in the query. Block 508 also identifies a hyperedge as being directed when it represents a one-sided outer join operation in the query. In addition, block 508 identifies a hyperedge as being bi-directed when it represents a full outer join operation in the query. Block 510 represents the output of the hypergraph for further processing.

GENERATION OF REQUIRED SETS

FIG. 6 is a flowchart illustrating the method of generating required sets for SQL queries according to the present invention. This procedure enumerates all the required sets (p) that identify constraints on associative re-orderings of the relations in the query Q. Required sets are generated only for left, right, and full outer joins.

This procedure uses the following notation:

T=T₁ ·T_(r) is an association tree of G_(leaves)(T), wherein T₁ is a left sub-tree of T and T_(r) is a right sub-tree of T.

E_(T) is a set of (hyper)edges in G that connects leaves of T₁ with leaves of T_(r). e is an (hyper)edge in E_(T).

Block 602 represents the input of the query hypergraph. Blocks 604-608 together form a loop that increments k=2, 3, . . . ,|E|. Block 606 represents the creation of a required set for every (bi)directed hyperedge e_(k) in E, wherein the required set (p) comprises {V₁,V_(r) }, sch(p) is a set of real attributes, V₁ =sch(p)∩T₁, and V_(r) =sch(p)∩T_(r), so that the required set (p) contains relations that must be in T₁ and T_(r) before T₁ and T_(r) can be combined. Block 608 represents the increment to the next k in the loop. Once the loop is completed, block 610 represents the output of the required sets.

GENERATION OF CONFLICT SETS

FIG. 7 is a flowchart illustrating the method of generating conflict sets for SQL queries according to the present invention. An association tree specifies an order in which relations are to be combined, but does not provide operators to apply at each step. In order to specify the operators, conflict sets are required for join and directed outer join (hyper)edges. Conflict sets are computed once from the query hypergraph and then are used for all association trees.

Block 702 represents the input of the query hypergraph. Blocks 704-708 together form a loop that increments k=2, 3, . . . ,|E|. Block 706 represents that, for each k, conf(e_(k)) is created for each join and directed hyperedge e_(k) in E. For a join edge e_(k), the set of closest conflicting outer join, denoted ccoj(e_(k)), is defined as follows: ##EQU26##

There can be at most one closest conflicting outer join (hyper)edge in ccoj(e_(k)). Also, if removing a set of join edges e₀,e₁, . . . ,e_(n) from G disconnects G into two connected graphs, then all these join edges have the same closest conflicting outer join (hyper)edge.

The set of conflicting (hyper)edges of undirected or directed (hyper)edge, e_(k), is denoted by conf(e_(k)). If e_(k) is directed, then the conflict set is defined as: ##EQU27## If e_(k) is undirected and ccoj(e_(k))=φ, then the conflict set is defined as: ##EQU28## If e_(k) is undirected and ccoj(e_(k))={e}, then the conflict set is defined as:

    conf(e.sub.k)={e}∪conf(e.sub.k)

In the above identities, ##EQU29## is a join edge or a left/right outer join (hyper)edge.

Intuitively, if (hyper)edge e₁ belongs to conf(e_(k)) then the operator corresponding to e₁ cannot be a descendant of the operator for e_(k) without using the MGOJ operator.

Block 708 represents the increment to the next k in the loop. Once the loop is completed, block 710 represents the output of the conflict sets conf(e_(k)).

GENERATION OF PRESERVED SETS

FIG. 8 is a flowchart illustrating the method of generating preserved sets for SQL queries according to the present invention. As indicated above, an association tree specifies an order in which relations are to be combined, but does not provide operators to apply at each step. In order to specify the operators, preserved sets are required for directed and bi-directed (hyper)edges. Like conflict sets, preserved sets are computed once from the query hypergraph and then are used for all association trees.

Block 802 represents the input of the query hypergraph. Blocks 804-808 together form a loop that increments k=2,3, . . . ,|E|. Block 806 represents that, for each k, the preserved set pres(e_(k)) is created for each (bi)directed hyperedge e_(k) in E.

In generating the preserved set pres(e_(k)), the preserved relations are identified. These preserved relations comprise the contents of the preserved sets pres(e_(k)). In a left outer join, ##EQU30## which is the relation <R₁ R₂, V₁ V₂, E'>, where: ##EQU31## relation r₁ is the preserved relation and relation r₂ is the null supplying relation. A right outer join, ##EQU32## can similarly be defined, wherein r₁ is the null supplying relation and r₂ is the preserved relation. In a full outer join, ##EQU33## both r₁ and r₂ are considered the null supplying and preserved relations.

If a directed (hyper)edge e_(o) =V_(i) →V_(j), is removed from hypergraph G, then G partitions into two connected hypergraphs, e.g., G₁ containing V_(i) and G₂ containing V_(j). The set of preserved attributes of e₀, denoted by pres(e₀), is the union of attributes of relations in G₁. Similarly, let e₁ be a (hyper)edge in graph G. If (hyper)edge e_(o) =(V_(i) ←→V_(j)) is removed from hypergraph G, then G partitions into two connected hypergraphs, e.g., G₁ containing e₁ and G₂ not containing e₁. The set of preserved attributes of e_(o) with respect to e₁, denoted pres_(e).sbsb.1 (e₀), is the union of attributes in G₂. If e_(o), is a directed (hyper)edge, then pres_(e).sbsb.1 (e₀) is (e₀)

Block 808 represents the increment to the next k in the loop. Once the loop is completed, block 810 represents the output of the preserved sets.

GENERATION OF ASSOCIATION TREES

FIG. 9 is a flowchart illustrating the method of generating association trees for SQL queries according to the present invention. This procedure enumerates all the possible association trees for a given query, wherein the association trees are constructed bottom up, incrementally, from the given query hypergraph. At each step, two sub-trees are combined to obtain a larger tree. Two sub-trees are combined only if all the conditions specified in the definition of association tree are satisfied. Although not mentioned here, dynamic programming can be used to prune the space of alternatives.

This procedure uses the following notation:

T=T₁ ·T_(r) is an association tree of G_(leaves)(T), wherein T₁ is a left sub-tree of T and T_(r) is a right sub-tree of T.

E_(T) is a set of (hyper)edges in G that connects leaves of T₁ with leaves of T_(r).

e is an (hyper)edge in E_(T).

Block 902 represents the input of the query hypergraph G=(V,E) and the required sets for the query. Block 904 represents the creation of a one leaf tree for each relation referenced in G. Blocks 906-910 together form a loop that increments k=2,3, . . . ,|V|. Block 908 represents that, for each k, sub-trees T₁ and T_(r) are combined to form T=T₁ ·T_(r) provided the following criteria are met:

1. leaves(T₁)∩leaves(T_(r))=φ

2. |leaves(T₁)|+|leaves(T_(r))|=k

3. G|_(leaves)(T.sbsb.1.sub.)∪leaves(T.sbsb.1.sub.) is connected.

4. ∀e=<V₁,V₂ >εE_(T), either V₁ .OR right.leaves(T₁) and V₂ .OR right.leaves(T_(r)), or V₂ .OR right.leaves(T₁) and V₁ .OR right.leaves(T_(r)).

Block 910 represents the increment to the next k in the loop. Once the loop is completed, block 912 represents the output of those trees having |V|leaves, i.e., all possible association trees for G.

CONFLICT-FREE OPERATOR ASSIGNMENT

FIGS. 10A, 10B, and 10C together are a flowchart illustrating the method of performing conflict-free operator assignments for the association trees according to the present invention. The conflict-free operator assignment attempts to generate an operator tree for a given association tree by assigning joins, outer joins and full outer joins to the interior nodes of the association tree. If it is not possible to generate an operator tree without using the MGOJ operator, then an empty operator tree is returned.

This procedure uses the following notation:

T=T₁ ·T_(r), wherein T₁ is a left sub-tree of T and T_(r) is a right sub-tree of T.

T'=T₁ ·T_(r), wherein · is the operator selected as the root of T.

E_(T) is the set of (hyper)edges in G that connects leaves of T₁ with leaves of T_(r) and e₀ =<V₁,V₂ >εE_(T).

C_(s) ={e|eεconf(e₀) and eεG|_(leaves)(T) }

Block 1002 represents the input of an association tree T and conflict sets C_(s) of a query graph G.

Block 1004 is a decision block that determines if p is a full outer join: ##EQU34## If so, then p_(v).sbsb.1_(v).sbsb.2 is the only hyperedge that connects leaves(T₁) with leaves(T_(r)). If so, block 1006 sets: ##EQU35## which is equivalent to: ##EQU36## and control then transfers to "A". If not, control transfers to block 1008.

Block 1008 is a decision block that determines if p is a one-sided outer join: ##EQU37## If so, then p_(v).sbsb.1_(v).sbsb.2 is the only hyperedge that connects leaves(T₁) with leaves(T_(r)). If not, control transfers to "B"; otherwise, control transfers to block 1010. Block 1010 is a decision block that determines if C_(s) =φ. If so, then block 1012 sets: ##EQU38## respectively, which is equivalent to: ##EQU39## respectively, and control then transfers to "A". If not, then block 1014 sets T'=φ and block 1016 returns the empty operator tree.

At "B", block 1018 is a decision block that determines if C_(s) =φ. If so, then block 1020 sets: ##EQU40## which is equivalent to: ##EQU41## and control then transfers to "A". If not, then block 1022 sets T'=φ and block 1024 returns the empty operator tree.

At "A", block 1026 recursively invokes the CONFLICT-FREE OPERATOR ASSIGNMENT procedure using sub-tree T₁ and query hypergraph G, and then block 1028 recursively invokes the CONFLICT-FREE OPERATOR ASSIGNMENT procedure using sub-tree T_(r) and query hypergraph G. Block 1030 represents the output of an operator tree T' for association tree T.

GENERAL OPERATOR ASSIGNMENT

FIGS. 11A, 11B, and 11C together are a flowchart illustrating the method of performing general operator assignments for the association trees according to the present invention. The conflict-free operator assignment does not generate operator trees for all possible association trees of a given query. Therefore, the general operator assignment is presented that can generate an operator tree for any association tree by employing MGOJ operators and preserved sets.

This procedure uses the following notation:

T=T₁ ·T_(r), wherein T₁ is a left sub-tree of T and T_(r) is a right sub-tree of T.

T'=T₁ ·T_(r), wherein · is the operator selected as the root of T.

E_(T) is the set of (hyper)edges in G that connects leaves of T₁ with leaves of T_(r) and e₀ =<V₁,V₂ >εE_(T).

C_(s) ={e|eεconf(e₀) and eεG|_(leaves)(T) }

Block 1102 represents the input of an association tree and conflict sets C_(s) of a query graph G.

Block 1104 is a decision block that determines if p is a full outer join: ##EQU42## If so, then p_(v).sbsb.1_(v).sbsb.2 is the only hyperedge that connects leaves(T₁) with leaves(T_(r)). If so, block 1106 sets: ##EQU43## which is equivalent to: ##EQU44## and control then transfers to "A". If not, control transfers to block 1108.

Block 1108 is a decision block that determines if p is a one-sided outer join: ##EQU45## If so, then p_(v).sbsb.1_(v).sbsb.2 is the only hyperedge that connects leaves(T₁) with leaves(T_(r)). If not, control transfers to "B"; otherwise, control transfers to block 1110.

Block 1110 is a decision block that determines if C_(s) =φ. If so, then block 1112 sets: ##EQU46## respectively, which is equivalent to: ##EQU47## respectively, and control then transfers to "A". If not, then block 1114 sets: ##EQU48## where eεE_(T) and block 1116 returns.

At "B", block 1118 is a decision block that determines if C_(s) =φ. If so, then block 1120 sets: ##EQU49## which is equivalent to: ##EQU50## and control then transfers to "A". If not, then block 1122 sets: ##EQU51## where eεE_(T), and block 1124 returns.

At "A", block 1126 recursively invokes the GENERAL OPERATOR ASSIGNMENT procedure using sub-tree T₁ and query hypergraph G, and then block 1128 recursively invokes the GENERAL OPERATOR ASSIGNMENT procedure using sub-tree T_(r) and query hypergraph G. Block 1130 represents the output of an operator tree T' for association tree T.

EXAMPLE SOL QUERY OPTIMIZATION

In one example according to the present invention, consider a query: ##EQU52## where X_(i) =<R_(i),V_(i),E_(i) >, 1≦i≦4, is a base relation and p_(i),j is the predicate between relations X_(i) and X_(j). The following association trees of Q₁ are generated by the ASSOCIATION TREES procedure:

(X₁ ·(X₂ ·X₃))·X₄

((X₁ ·X₃)·X₂)·X₄

X₁ ·((X₂ ·X₃)·X₄)

The CONFLICT-FREE OPERATOR ASSIGNMENT procedure generates the following operator trees: ##EQU53##

The GENERAL OPERATOR ASSIGNMENT procedure generates the following operator trees: ##EQU54##

IDENTITIES FOR JOIN, OUTER AND FULL OUTER ASSOCIATION

Let X_(i) =<R_(i),V_(i),E_(i) >, where 1≦i≦3, be an expression, p_(i),j denotes the predicate(s) between expressions X_(i) and X_(j), then: ##EQU55## where R_(X').sbsb.2 .OR right.R₂ and R_(X').sbsb.3 .OR right.R₃. In addition to the above identities, the following identities are also used: ##EQU56##

CONCLUSION

This concludes the description of the preferred embodiment of the invention. The following describes some alternative embodiments for accomplishing the present invention. For example, any type of computer, such as a mainframe, minicomputer, or personal computer, could be used with the present invention. In addition, any software program adhering (either partially or entirely) to the SQL language could benefit from the present invention.

In summary, the present invention discloses a method for reordering complex SQL queries containing joins, outer and full outer joins. The method first translates the query into a hypergraph representation. Required sets, conflict sets and preserved sets are then generated for the query hypergraph. Using the required sets, a plurality of plans are enumerated, wherein the plans represent associative re-orderings of relations in the query. SQL operators are selectively assigned to each of the enumerated plans using the conflict sets and/or preserved sets, so that the results from the plans are identical to the original query. A novel Modified General Outer Join (MGOJ) operator may be assigned to the root of a sub-tree, wherein the MGOJ operator is a compensation operator. The operator assignment is performed recursively for the root of each sub-tree in the plan. One of the enumerated plans (generally the most optimal) is then selected for execution.

The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. 

What is claimed is:
 1. A method of reordering a query in a computer having a memory, the query being used by the computer to direct information retrieval from a relational database stored in an electronic storage device coupled to the computer, the method comprising the steps of:(a) accepting the query into the memory of the computer; (b) enumerating one or more plans for the query in the memory of the computer, wherein the enumerated plans represent associative re-orderings of relations in the query; and (c) selectively assigning a Modified General Outer Join (MGOJ) operator at a root of one or more of the enumerated plans.
 2. The method of claim 1, further comprising the step of:(d) selecting one of the enumerated plans for execution by the computer according to a predefined criteria for the retrieval of the information from the relational database stored in the electronic storage device.
 3. The method of claim 1, wherein the MGOJ operator is a compensation operator.
 4. The method of claim 1, wherein the MGOJ operator comprises a relation <R₁ R₂,V₁ V2,E'> in an expression comprising:

    r.sub.1 MGOJ  p,X.sub.1, . . . ,X.sub.n,Y.sub.1, . . . ,Y.sub.n' !r.sub.2

for relations r₁ and r₂ that preserves relations X_(i) and Y_(j), wherein: r₁ =<R₁,V₁,E₁ >, wherein R₁ is a non-empty set of real attributes, V₁ is a non-empty set of virtual attributes, and E₁ is a set of tuples that comprise the relation r₁, r₂ =<R₂,V₂,E₂ >, wherein R₂ is a non-empty set of real attributes, V₂ is a non-empty set of virtual attributes, and E₂ is a set of tuples that comprise the relation r₂, R₁ ∩R₂ =φ, V₁ ∩V₂ =φ, X_(i) =<R_(x).sbsb.i,V_(x).sbsb.i,E_(x).sbsb.i >for 1≦i≦n, Y_(j) =<R_(y).sbsb.j,V_(y).sbsb.j,E_(y).sbsb.j > for 1≦j≦n', X_(i) and Y_(j) are relations such that R_(x).sbsb.i ∩R_(x).sbsb.k =V_(x).sbsb.i ∩V_(x).sbsb.k =φ, R_(y).sbsb.j ∩R_(y).sbsb.s =V_(y).sbsb.j ∩V_(y).sbsb.s =φ, R_(x).sbsb.i .OR right.R₁ and R_(y).sbsb.j .OR right.R₂, where i≠k, j≠s, 1≦i, k≦n, and 1≦j,s≦n', ##EQU57## p_(i),j is a predicate for ##EQU58## ·ε{, →, ←, ←→} defines operators for the query, denotes a join operation, → denotes a left outer join operation, ← denotes a right outer join operation, ←→ denotes a full outer join operation, and π^(c) denotes a projection operation.
 5. The method of claim 4, wherein the assigning step comprises replacing: ##EQU59## with: ##EQU60##
 6. The method of claim 4, wherein the assigning step comprises replacing: ##EQU61## with: ##EQU62##
 7. The method of claim 4, wherein the assigning step comprises replacing: ##EQU63##
 8. The method of claim 4, wherein the assigning step comprises replacing: ##EQU64##
 9. An apparatus for reordering a query, comprising:(a) a computer having a memory, the query being used by the computer to direct information retrieval from a relational database stored in an electronic storage device coupled to the computer, the computer further comprising:(1) means for accepting the query into the memory of the computer; (2) means for enumerating one or more plans for the query in the memory of the computer, wherein the enumerated plans represent associative re-orderings of relations in the query; and (3) means for selectively assigning a Modified General Outer Join (MGOJ) operator at a root of one or more of the enumerated plans.
 10. The apparatus of claim 9, wherein the computer further comprises:(4) means for selecting one of the enumerated plans for execution by the computer according to a predefined criteria for the retrieval of the information from the relational database stored in the electronic storage device.
 11. The apparatus of claim 9, wherein the MGOJ operator is a compensation operator.
 12. The apparatus of claim 9, wherein the MGOJ operator comprises a relation <R_(i) R₂,V₁ V2,E'> in an expression comprising:

    r.sub.1 MGOJ  p,X.sub.1, . . . ,X.sub.n,Y.sub.1, . . . ,Y.sub.n' !r.sub.2

for relations r₁ and r₂ that preserves relations X_(i) and Y_(j), wherein: r₁ =<R₁,V₁,E₁ >, wherein R₁ is a non-empty set of real attributes, V₁ is a non-empty set of virtual attributes, and E₁ is a set of tuples that comprise the relation r₁, r₂ =<R₂,V₂,E₂ >, wherein R₂ is a non-empty set of real attributes, V₂ is a non-empty set of virtual attributes, and E₂ is a set of tuples that comprise the relation r₂, R₁ ∩R₂ =φ, V₁ ∩V₂ =φ, X_(i) =<R_(x).sbsb.i,V_(x).sbsb.1,E_(x).sbsb.i > for 1≦i≦n, Y_(j) =<R_(y).sbsb.j,V_(y).sbsb.j,E_(y).sbsb.j > for 1≦j≦n', X_(i) and Y_(j) are relations such that R_(x).sbsb.i ∩R_(x).sbsb.k =V_(x).sbsb.i ∩V_(x).sbsb.k =φ, R_(y).sbsb.j ∩R_(y).sbsb.s =V_(y).sbsb.j ∩V_(y).sbsb.s =φ, R_(x).sbsb.i .OR right.R₁ and R_(y).sbsb.j .OR right.R₂, where i≠k, j≠s, 1≦i,k≦n, and 1≦j,s≦n', ##EQU65## p_(i),j is a predicate for ##EQU66## ·ε{, →, ←, ←→} defines operators for the query, denotes a join operation, → denotes a left outer join operation, ← denotes a right outer join operation, ←→ denotes a full outer join operation, and π^(c) denotes a projection operation.
 13. The apparatus of claim 12, wherein the means for assigning comprises means for replacing: ##EQU67##
 14. The apparatus of claim 12, wherein the means for assigning comprises means for replacing: ##EQU68##
 15. The apparatus of claim 12, wherein the means for assigning comprises means for replacing: ##EQU69##
 16. The apparatus of claim 12, wherein the means for assigning comprises means for replacing: ##EQU70##
 17. A program storage device, readable by a computer, tangibly embodying a program of instructions executable by the computer to perform method steps reordering a query in a computer having a memory, the query being used by the computer to direct information retrieval from a relational database stored in an electronic storage device coupled to the computer, the method comprising the steps of:(a) accepting the query into the memory of the computer; (b) enumerating one or more plans for the query in the memory of the computer, wherein the enumerated plans represent associative re-orderings of relations in the query; and (c) selectively assigning a Modified General Outer Join (MGOJ) operator at a root of one or more of the enumerated plans.
 18. The method of claim 17, further comprising the step of:(d) selecting one of the enumerated plans for execution by the computer according to a predefined criteria for the retrieval of the information from the relational database stored in the electronic storage device.
 19. The method of claim 17, wherein the MGOJ operator is a compensation operator.
 20. The method of claim 17, wherein the MGOJ operator comprises a relation <R₁ R₂,V₁ V₂,E'> in an expression comprising:

    r.sub.1 MGOJ  p,X.sub.1, . . . ,X.sub.n,Y.sub.1, . . . ,Y.sub.n' !r.sub.2

for relations r₁ and r₂ that preserves relations X_(i) and Y_(j), wherein: r₁ =<R₁,V₁,E₁ >, wherein R₁ is a non-empty set of real attributes, V₁ is a non-empty set of virtual attributes, and E₁ is a set of tuples that comprise the relation r₁, r₂ =<R₂,V₂,E₂ >, wherein R₂ is a non-empty set of real attributes, V.sub. is a non-empty set of virtual attributes, and E₂ is a set of tuples that comprise the relation r₂, R₁ ∩R₂ =φ, V₁ ∩V₂ =φ, X_(i) =<R_(x).sbsb.i,V_(x).sbsb.i,E_(x).sbsb.i > for 1≦i≦n, Y_(j) =<R_(y).sbsb.j,V_(y).sbsb.j,E_(y).sbsb.j > for 1≦j≦n', X_(i) and Y_(j) are relations such that R_(x).sbsb.i ∩R_(x).sbsb.k =V_(x).sbsb.i ∩V_(x).sbsb.k =φ, R_(y).sbsb.j ∩R_(y).sbsb.s =V_(y).sbsb.j ∩V_(y).sbsb.s =φ, R_(x).sbsb.i .OR right.R₁ and R_(y).sbsb.j .OR right.R₂, where i≠k, j≠s, 1≦i,k≦n, and 1≦j,s≦n', ##EQU71## p_(i),j is a predicate for ##EQU72## ·ε{, →, ←, ←→} defines operators for the query, denotes a join operation, → denotes a left outer join operation, ← denotes a right outer join operation, ←→ denotes a full outer join operation, and π^(c) denotes a projection operation.
 21. The method of claim 20, wherein the assigning step comprises replacing: ##EQU73##
 22. The method of claim 20, wherein the assigning step comprises replacing: ##EQU74##
 23. The method of claim 20, wherein the assigning step comprises replacing: ##EQU75##
 24. The method of claim 20, wherein the assigning step comprises replacing: ##EQU76## 